Kähler 曲面的条件均匀化
Conditional Uniformization of Kähler Surfaces
AI总结：
本文证明在非负 Ricci 和二次正交双截曲率条件下，无穷远处单连通的完备非紧 Kähler 曲面可缩且同胚于 $\mathbb{R}^4$，并改进了条件均匀化定理，同时给出 $\mathbb{C}^n$ 上不变度量的穷竭函数与核估计。
AI中文摘要：
我们证明了，具有非负 Ricci 曲率和非负二次正交双截曲率的完备非紧 Kähler 曲面，若在无穷远处单连通，则是可缩的，从而同胚于 $\mathbb{R}^4$。在正双截曲率下，这去除了 Datar--Pingali--Seshadri 条件均匀化定理中的可缩性假设：强 Stein 性和无穷远处单连通性足以将该曲面双全纯地等同于 $\mathbb{C}^2$。我们还推导了 $\mathbb{C}^n$ 上具有非负双截曲率的完备 $U(n)$-不变 Kähler 度量的有界梯度严格多次调和穷竭函数以及一致全纯核估计。
英文摘要：
We prove that a complete noncompact Kähler surface with nonnegative Ricci and nonnegative quadratic orthogonal bisectional curvature is contractible, and hence homeomorphic to $\mathbb{R}^4$, if it is simply connected at infinity. Under positive bisectional curvature, this removes the contractibility assumption from the conditional uniformization theorem of Datar--Pingali--Seshadri: strong Steinness and simple connectivity at infinity suffice to identify the surface biholomorphically with $\mathbb{C}^2$. We also derive bounded-gradient strictly plurisubharmonic exhaustions and uniform holomorphic kernel estimates for complete $U(n)$-invariant Kähler metrics on $\mathbb{C}^n$ with nonnegative bisectional curvature.
